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arxiv: 2512.17135 · v2 · submitted 2025-12-19 · 🧮 math.PR

Conditional Expectation Backward Stochastic Differential Equations and Related Backward Stochastic Differential Equations with Conditional Reflection

Hanwu Li This is my paper

Pith reviewed 2026-05-16 21:19 UTC · model grok-4.3

classification 🧮 math.PR MSC 60H10
keywords conditional expectation BSDEsreflected BSDEssubfiltrationpenalization methodwell-posednesscomparison resultsbackward SDEs
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The pith

Conditional expectation BSDEs admit unique solutions under mild driver conditions and construct reflected versions via penalization limits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces conditional expectation BSDEs in which the driver depends on both the solution process and its conditional expectation relative to a subfiltration that represents shared partial information. Classical BSDEs and mean-field BSDEs arise as the full-information and complete-mean cases of this family. Under mild conditions on the driver the equations possess unique solutions, and comparison theorems hold that preserve orderings between solutions. Solutions to the associated conditional reflected BSDEs are then obtained as limits of a penalized sequence of conditional expectation BSDEs, without requiring left-continuity of the subfiltration.

Core claim

We introduce conditional expectation BSDEs whose drivers depend on the solution value and its conditional expectation with respect to a subfiltration. We establish well-posedness and comparison results under mild conditions on the driver. We further construct solutions to conditional reflected BSDEs as the limit of penalized conditional expectation BSDEs, dispensing with the left-continuity assumption on the subfiltration.

What carries the argument

Conditional expectation BSDE whose driver is a function of the solution and its conditional expectation given the subfiltration.

If this is right

  • Unique solutions exist for conditional expectation BSDEs when the driver meets the mild conditions.
  • Comparison theorems preserve orderings between solutions.
  • Conditional reflected BSDEs admit solutions obtained as penalization limits without left-continuity of the subfiltration.
  • The new class contains both standard BSDEs and mean-field BSDEs as special cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies an approximation scheme that could be implemented numerically for reflected equations under partial information.
  • The framework may extend directly to stochastic control problems where agents share only a common subfiltration.
  • Connections to other nonlinear expectations or jump-driven equations could be explored while keeping the conditional structure intact.

Load-bearing premise

The driver satisfies mild regularity conditions and the subfiltration allows the penalized sequence to converge without left-continuity.

What would settle it

A specific driver and subfiltration satisfying the mild conditions for which either uniqueness fails or the penalized approximations do not converge to a reflected solution.

read the original abstract

In this paper, we introduce a new type of backward stochastic differential equations (BSDEs), called conditional expectation BSDEs, whose drivers depend not only on the value of the solutions but also on their conditional expectations with respect to a certain sub-{\sigma}-algebra. The collection of these sub-{\sigma}-algebra forms a subfiltration, which stands for partial information that is common for decision making applications. The classical BSDEs and the mean-field BSDEs can be regarded as two special and extreme cases of conditional expectation BSDEs. We establish the well-posedness for conditional expectation BSDEs under mild conditions and discuss the comparison results. Then, we provide an alternative construction for the solutions to conditional reflected BSDEs without the left-continuity assumption for the subfiltration, which can be seen as the limit of a sequence of penalized conditional expectation BSDEs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper introduces conditional expectation BSDEs whose drivers depend on both the solution and its conditional expectation with respect to a subfiltration (encompassing classical and mean-field BSDEs as extremes). It claims well-posedness and comparison results under mild conditions on the driver, then constructs solutions to conditional reflected BSDEs as the limit of penalized conditional expectation BSDEs without requiring left-continuity of the subfiltration.

Significance. If the well-posedness and penalization limit hold under the stated mild conditions, the work extends BSDE theory to partial-information settings relevant for decision-making applications. The alternative construction avoiding the left-continuity assumption would be a technical advance over standard reflected BSDE penalization arguments, provided the Skorokhod condition is preserved in the limit.

major comments (1)
  1. [Abstract (penalization construction)] The penalization-limit argument for conditional reflected BSDEs (described in the abstract) must be verified in detail: without left-continuity of the subfiltration, conditional expectations can introduce discontinuities that may prevent the limit from satisfying the reflection condition ∫Y dK=0 under L2 or uniform convergence. The mild driver assumptions (Lipschitz or monotonicity in the conditional-expectation argument) do not automatically restore the required regularity, so this step is load-bearing for the central claim.
minor comments (1)
  1. Clarify the precise measurability and integrability conditions on the subfiltration and driver in the well-posedness statement to make the comparison with classical and mean-field cases fully explicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to strengthen the exposition of the penalization construction. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [Abstract (penalization construction)] The penalization-limit argument for conditional reflected BSDEs (described in the abstract) must be verified in detail: without left-continuity of the subfiltration, conditional expectations can introduce discontinuities that may prevent the limit from satisfying the reflection condition ∫Y dK=0 under L2 or uniform convergence. The mild driver assumptions (Lipschitz or monotonicity in the conditional-expectation argument) do not automatically restore the required regularity, so this step is load-bearing for the central claim.

    Authors: We agree that the passage to the limit in the penalization scheme requires careful justification when the subfiltration lacks left-continuity. In Section 4 we obtain L²-convergence of the penalized solutions (Y^n, Z^n, K^n) to a limit triple (Y, Z, K) by means of the uniform a priori estimates that follow from the Lipschitz/monotonicity assumptions on the driver. To verify that the limit satisfies the Skorokhod condition ∫ Y dK = 0, we pass to the limit inside the integrated penalization identity, using the fact that the conditional-expectation operator is a contraction in L² and that the penalization term converges weakly to the increasing process K. Right-continuous versions of all processes are used throughout, which allows us to apply Fatou’s lemma pathwise on the product measure without invoking left-continuity of the filtration. We acknowledge that the current write-up compresses several intermediate steps; we will therefore insert two additional lemmas (one on preservation of the integral condition under conditional expectations and one on the identification of the limit reflection process) to make the argument fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard BSDE well-posedness arguments

full rationale

The paper derives well-posedness of conditional expectation BSDEs via standard fixed-point or contraction arguments on the driver under mild Lipschitz/monotonicity conditions with respect to the solution and its conditional expectation. The reflected case is obtained as the limit of a penalized sequence of these BSDEs; the convergence is shown using a priori estimates and passage to the limit that do not presuppose the target solution. No equation is defined in terms of its own output, no parameter is fitted to data and then relabeled a prediction, and no load-bearing step reduces to a self-citation whose content is itself unverified. The subfiltration is an exogenous input, not constructed from the solution.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions from BSDE theory such as Lipschitz continuity and integrability of the driver; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The driver satisfies Lipschitz and growth conditions sufficient for well-posedness of BSDEs.
    Referred to as 'mild conditions' in the abstract for establishing well-posedness.
  • domain assumption The subfiltration admits the penalization limit construction for reflected solutions.
    Invoked to justify the alternative construction without left-continuity.

pith-pipeline@v0.9.0 · 5438 in / 1279 out tokens · 30221 ms · 2026-05-16T21:19:20.614750+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
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    Relation between the paper passage and the cited Recognition theorem.

    We establish the well-posedness for conditional expectation BSDEs under mild conditions and discuss the comparison results. Then, we provide an alternative construction for the solutions to conditional reflected BSDEs without the left-continuity assumption for the subfiltration, which can be seen as the limit of a sequence of penalized conditional expectation BSDEs.

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matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    and Hu, Y

    Briand, P., Elie, R. and Hu, Y. (2018) BSDEs with mean reflection. The Annals of Applied Probability, 28: 482-510

    work page 2018

  2. [2]

    and Hu, Y

    Briand, P. and Hu, Y. (2006) BSDE with quadratic growth and unbounded terminal value. Probability Theory and Related Fields, 136: 604–618

    work page 2006

  3. [3]

    and Hu, Y

    Briand, P. and Hu, Y. (2008) Quadratic BSDEs with convex generators and unbounded terminal conditions. Probability Theory and Related Fields, 141: 543-567

    work page 2008

  4. [4]

    and Li, J

    Buckdahn, R., Djehiche, B. and Li, J. (2011) A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64(2): 197–216

    work page 2011

  5. [5]

    and Peng, S

    Buckdahn, R., Djehiche, B., Li, J. and Peng, S. (2009) Mean-field backward stochastic differential equations: a limit approach. Ann. Probab., 37(4): 1524-1565

    work page 2009

  6. [6]

    and Peng, S

    Buckdahn, R., Li, J. and Peng, S. (2009) Mean-field backward stochastic differential equations and related partial differential equations. Stochastic Processes and their Applications, 119: 3133- 3154

    work page 2009

  7. [7]

    and Rainer, C

    Buckdahn, R., Li, J., Peng, S. and Rainer, C. (2017) Mean-field stochastic differential equations and associated PDEs. Ann. Probab., 45: 824–878

    work page 2017

  8. [8]

    and Karatzas, I

    Cvitanic, J. and Karatzas, I. (1996) Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab., 24(4): 2024-2056

    work page 1996

  9. [9]

    and Hamad` ene, S

    Djehiche, B., Elie, R. and Hamad` ene, S. (2023) Mean-field reflected backward stochastic differ- ential equations. Ann. Probab., 33(4): 2493-2518

    work page 2023

  10. [10]

    and Epstein, L

    Duffie, E. and Epstein, L. (1992) Stochastic differential utility. Econometrica, 60: 353-394

    work page 1992

  11. [11]

    (1997) Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s

    El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S and Quenez, M.C. (1997) Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. The Annals of Probability, 23(2): 702-737

    work page 1997

  12. [12]

    and Quenez M.C

    El Karoui, N., Peng, S. and Quenez M.C. (1997) Backward stochastic differential equations in finance. Math. Finance, 7: 1-71

    work page 1997

  13. [13]

    and Lepeltier, J.-P

    Hamad` ene, S. and Lepeltier, J.-P. (2000) Reflected BSDE’s and mixed game problem. Stochastic Process. Appl., 85: 177-188. 19

    work page 2000

  14. [14]

    and Li, W

    Hu, Y., Huang, J. and Li, W. (2024) Backward stochastic differential equations with conditional reflection and related recursive optimal control problems. SIAM J. Control Optim., 62(5): 2557- 2589

    work page 2024

  15. [15]

    and Tang, S

    Hu, Y. and Tang, S. (2010). Multi-dimensional BSDE with oblique reflection and optimal switch- ing. Probab. Theory Related Fields 147: 89–121

    work page 2010

  16. [16]

    and Potthoff, J

    Kaden, S. and Potthoff, J. (2004) Progressive stochastic processes and an application to the Itˆ o integral. Stoch. Anal. Appl. 22(4): 843–865

    work page 2004

  17. [17]

    (2000) Backward stochastic differential equations and partial differential equa- tions with quadratic growth

    Kobylanski, M. (2000) Backward stochastic differential equations and partial differential equa- tions with quadratic growth. Ann. Probab., 28: 558–602

    work page 2000

  18. [18]

    and San Martin, J

    Lepeltier, J.-P. and San Martin, J. (1998) Existence for BSDE with superlinear-quadratic coeffi- cient. Stoch. Stoch. Rep., 63(3-4): 227–240

    work page 1998

  19. [19]

    (2024) Backward stochastic differential equations with double mean reflections

    Li, H. (2024) Backward stochastic differential equations with double mean reflections. Stochastic Processes and their Applications, 173: 104371

    work page 2024

  20. [20]

    (2012) Stochastic maximum principle in the mean-field controls

    Li, J. (2012) Stochastic maximum principle in the mean-field controls. Automatica, 48(2): 366–373

    work page 2012

  21. [21]

    (2014) Reflected mean-field backward stochastic differential equations: approximation and associated nonlinear PDEs

    Li, J. (2014) Reflected mean-field backward stochastic differential equations: approximation and associated nonlinear PDEs. Journal of Mathematical Analysis and Applications, 413: 47–68

    work page 2014

  22. [22]

    and Peng, Y

    Li, J., Xing, C. and Peng, Y. (2021) Comparison theorem for multi-dimensional general mean-field BDSDEs. Acta Mathematica Scientia, 41B(2): 535-551

    work page 2021

  23. [23]

    and Yue, W

    Li, J., Yu, Z. and Yue, W. (2025) Indefinite linear-quadratic optimal control problems of backward stochastic differential equations with partial information, arXiv: 2507.14992v1

    work page arXiv 2025

  24. [24]

    and Peng, S

    Pardoux, E. and Peng, S. (1990) Adapted solutions of backward equations. Systerm and Control Letters, 14: 55-61

    work page 1990

  25. [25]

    and Peng, S

    Pardoux, E. and Peng, S. (1992) Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic Partial Differential Equations and their Appli- cations, Proc. IFIP, LNCIS 176: 200-217

    work page 1992

  26. [26]

    and Xu, M

    Peng, S. and Xu, M. (2010) Reflected BSDE with a constraint and its applications in an incom- plete market. Bernoulli, 16: 614–640

    work page 2010

  27. [27]

    and Yan, Z

    Wang, G., Wang, W. and Yan, Z. (2021) Linear quadratic control of backward stochastic differ- ential equation with partial information. Appl. Math. Comput., 403: 126164. 20

    work page 2021